Tag Archives: proof

On August 6, Vinay Deolalikar, a researcher at HP Labs in Palo Alto, released a paper with a proposed proof for the open problem P != NP. Somehow, news of this proof made it to the mainstream press with headlines proclaiming that the problem was solved. Problem was, the computational complexity community not only had not had to time to review the proof, but viewed it with great skepticism.

The significance of this is that proving P != NP is a grand challenge problem. You will hear that if it turned out the other way, that is, P = NP, then all known cryptographic systems would become easily breakable. On the other hand, it would mean that many verification and optimization problems would suddenly become tractable. More troubling, P=NP would also would mean that Skynet is just around the corner. Most researchers already believe that P != NP, so there is really no immediate effect from producing a valid proof of P!+NP.

The P!+NP problem has been deemed so important to solve that whoever solves it either way will receive a $1M prize from the Clay Mathematics Institute. Before this happens, though, the proof would need to be generally accepted by the research community as being correct, a process that will probably take months, if not years.

An extraordinary thing happened in response to the release of the proof. A number of experts in the field dropped what they were doing and started poring over the proof in earnest. This is no small feat given that the proof is over 100 pages long. Pretty quickly, a consensus was reached that there were a number of serious flaws, although, as far as I can tell, no smoking gun yet. For his part, Deolalikar has not conceded defeat yet and is working on addressing all issues that were found.

The proof and mathematics involved are beyond my pay grade. But, the most interesting part of this episode for me is the sociological issues brought up. A group of the top experts in the world dropping everything to review a paper in a matter of days is highly unusual. Unprecedented is that fact that the discussion happened entirely out in the open in online forums. This is in contrast to the normal peer review process, which, for journals, can stretch to years, and is often done by only a few anonymous reviewers who generally don’t share their discussion outside the editorial board (or conference program committee as the case may be).

The discussion could be followed by anyone interested. Now, it is not clear that this is a scalable model as is: it is not possible to drop everything at a moments notice for every paper that comes along. This could have the negative effect of, the next time this happens due to the crying wolf effect. On the positive side, the openness of the process allowed the computational complexity to come together and get in sync on this problem. Many of the objections raised were “standard” objections. For example, the proof may prove other things that are known to be false. This is something that researchers routinely check for when reviewing a paper.

Researchers have an, essentially, agreed upon set of “barriers” to a proof of P!=NP. They also point to a number of results (theorems) that any valid proof would need to address. Any proof in the future (assuming Deolalikar’s doesn’t work out) would need to address these barriers. Putting all this discussion in the open, hopefully, means that proposed proofs in the future would be much higher quality ( P!=NP (and some P+NP) “proofs” are produced on a regular basis).

The other thing that is clear from this episode is that anyone who is going to receive the $1M prize needs the blessing of the computation complexity community. The most interesting barrier that I learned about is a theorem that essentially says “proofs of P!=NP are hard because P!+NP”. All of the other barriers also imply that any proof must be hard. No one is going to come up with a short, elegant, easy-to-understand proof for this. This means that a proof is highly unlikely to come from an outsider. Deolalikar is an outsider, but managed to meet enough of the community requirements to be taken seriously. While researchers acknowledge the contributions of outsiders, the nature of this problem makes it highly unlikely to come from an outsider.

In the (likely) event that Deolalikar’s proof ends up being incorrect, years from now, when a proof is produced, I think we will look back at this event and see that it was a step forward because of the openness of the discussion. Here’s hoping that this leads to positive improvements in how scientific review is done. I think we should be grateful to Deolalikar even if his proof doesn’t work out and even if it represents no new insight into the P!=NP problem.

For further reading:

bbc news (an easy to understand article blessed by an expert in the field)